On the robustness of the integrable trajectories of the control systems with limited control resources

The control system described by Urysohn type integral equation is considered where the system is nonlinear with respect to the phase vector and is affine with respect to the control vector. The control functions are chosen from the closed ball of the space $L_q\left(\Omega;\mathbb{R}^m\right),$ $q>1,$ with radius $r$ and centered at the origin. The trajectory of the system is defined as $p$-integrable multivariable function from the space $L_p\left(\Omega;\mathbb{R}^n\right),$ $\frac{1}{q}+\frac{1}{p}=1,$ satisfying the system's equation almost everywhere. It is shown that the system's trajectories are robust with respect to the fast consumption of the remaining control resource. Applying this result it is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the total control resource.